Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$ (...) a negative answer to one of the (open) questions posed in the final section of arxiv.org/abs/1506.04664 (namely, Question 6), provided we work, e.g., in a model of ZF where at least two measures with the above characteristics exist. As a matter of fact, we don't know whether the answer to the same question is negative in an arbitrary model of ZF.

Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$ I was hoping for your answer, Martin. I mentioned this thread to Georges Grekos, and he commented something like: "Since you put it in MO, Martin Sleziak will answer..." :-) Thank you so much! I must confess that I've had a hard time reading van Douwen's paper, and stopped just at Section 5, as it wasn't, and isn't yet, clear to me whether Constructions 5.2 and 5.3 in that very section give rise to different measures or not. The reason why I was so interested is that having more than one additive measure on ${\bf N}^+$ with the rigidity properties specified in the OP yields (...)